\(\large\mbox{Two random variable as a vector:}\)

\[ \begin{pmatrix} X \\ Y \end{pmatrix}\sim N(\mu, \Sigma), \mbox{ where } \mu=\begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix} \mbox{ and } \Sigma=\begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix} \]

1 Dependent

\[ \begin{pmatrix} X \\ Y \end{pmatrix}\sim N( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}) \]

\(\rho=0.3\)

plot_ly(data = dep, x = ~V1, y = ~V2)

\(\rho=0.5\)

plot_ly(data = dep, x = ~V1, y = ~V2)

\(\rho=0.8\)

plot_ly(data = dep, x = ~V1, y = ~V2)

2 Independent

\[ \begin{pmatrix} X \\ Y \end{pmatrix}\sim N( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}) \]

3 Independent(ind) vs Dependent(dep)

plot_ly(data = bvrnorm, x = ~V1, y = ~V2, color = ~V3)

4 New case

\[ \begin{pmatrix} X \\ Y \end{pmatrix}\sim N( \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix})\\ \]

Two new random variables: \(T = X+Y\) and \(S = X-Y\)

plot_ly(data = New_bvrnorm, x = ~U, y = ~V)